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Metamath Proof Explorer

Theorem eufsn2

Description: There is exactly one function into a singleton, assuming ax-pow and ax-un . Variant of eufsn . If existence is not needed, use mofsn or mofsn2 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024)

	Ref	Expression
Hypotheses	eufsn.1	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
	eufsn.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
Assertion	eufsn2	⊢ ( 𝜑 → ∃! 𝑓 𝑓 : 𝐴 ⟶ { 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		eufsn.1	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
2		eufsn.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3		snex	⊢ { 𝐵 } ∈ V
4		xpexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ { 𝐵 } ∈ V ) → ( 𝐴 × { 𝐵 } ) ∈ V )
5	2 3 4	sylancl	⊢ ( 𝜑 → ( 𝐴 × { 𝐵 } ) ∈ V )
6	1 5	eufsnlem	⊢ ( 𝜑 → ∃! 𝑓 𝑓 : 𝐴 ⟶ { 𝐵 } )